When he gives Aunt May the eggs he was supposed to bring earlier, he is clearly injured but Aunt May does not suggest that he is Spider-man. (At least that's what I remember)

However, after Gwen goes to his door, Aunt May asks why he didn't ask her out and he replies that he's no good for her. Then Aunt May says something about him being good - does she know he is Spider-man?

6 Answers
6

No, Aunt May didn't know that Peter was Spider-Man nor did she have any inklings. Rather, what's she seeing is a teenager stricken by grief from the passing of his uncle, which he believes to be his fault, finally coming to terms with his passing and trying to get back on track.

To Aunt May, Peter's just a teenager going through a rough time, getting into fights and dealing with the passing of his uncle. Him finally bring home the eggs is a sign to her that he's making an effort, instead of just forgetting everything like before.

I've just watched the film again for the 3rd or 4th time and I actually think she does, because her reaction to the news and then when he walks in afterwards, she tells him it'll be ok, as a reaction to what she saw

Aunt May doesn't know that Peter is spider-man. She thinks he is a just a teenager having a tough time, and when he finally get the eggs, to her its just a sign of him trying to make an effort to try to get back on a better note with her.